topos

For topoi in literary theory, see Literary topos. For topoi in rhetorical invention, see Inventio.

In mathematics, a topos (plural "topoi" or "toposes") is a type of category that behaves like the category of sheaves of sets on a topological space. For a discussion of the history of topos theory, see the article Background and genesis of topos theory.

Grothendieck topoi (topoi in geometry)

Since the introduction of sheaves into mathematics in the 1940s a major theme has been to study a space by studying sheaves on that space. This idea was expounded by Alexander Grothendieck by introducing the notion of a topos. The main utility of this notion is in the abundance of situations in mathematics where topological intuition is very effective but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the intuition. The greatest single success of this programmatic idea to date has been the introduction of the étale topos of a scheme.

Equivalent formulations

Let C be a category. A theorem of Giraud states that the following are equivalent:

There is a category D and an inclusion C $\hookrightarrow$ Presh(D) that admits a left adjoint.
C is the category of sheaves on a Grothendieck site.
C satisfies Giraud's axioms, below.

A category with these properties is called a "(Grothendieck) topos". Here Presh(D) denotes the category of contravariant functors from D to the category of sets; such a contravariant functor is frequently called a presheaf.

Giraud's axioms

Giraud's axioms for a category C are:

C has a small set of generators, and admits arbitrary colimits. Furthermore, colimits commute with base change.
Sums in C are disjoint. In other words, the fiber product of X and Y over their sum is the initial object in C.
All equivalence relations in C are effective.

The last axiom needs the most explanation. If X is an object of C, an equivalence relation R on X is a map R→X×X in C such that all the maps Hom(Y,R)→Hom(Y,X)×Hom(Y,X) are equivalence relations of sets. Since C has colimits we may form the coequalizer of the two maps R→X; call this X/R. The equivalence relation is effective if the canonical map

$R \to X \times_{X/R} X \,\!$

is an isomorphism.

Examples

Giraud's theorem already gives "sheaves on sites" as a complete list of examples. Note, however, that nonequivalent sites often give rise to equivalent topoi. As indicated in the introduction, sheaves on ordinary topological spaces motivate many of the basic definitions and results of topos theory.

The category of sets is an important special case: it plays the role of a point in topos theory. Indeed, a set may be thought of as a sheaf on a point.

More exotic examples, and the raison d'être of topos theory, come from algebraic geometry. To a scheme and even a stack one may associate an étale topos, an fppf topos, a Nisnevich topos...

Counterexamples

Topos theory is, in some sense, a generalization of classical point-set topology. One should therefore expect to see old and new instances of pathological behavior. For instance, there is an example due to Pierre Deligne of a nontrivial topos that has no points.

Geometric morphisms

If X and Y are topoi, a geometric morphism u:X→Y is a pair of adjoint functors (u^∗,u_∗) such that u^∗ preserves finite limits. Note that u^∗ automatically preserves colimits by virtue of having a right adjoint.

If X and Y are topological spaces and u is a continuous map between them, then the pullback and pushforward operations on sheaves yield a geometric morphism between the associated topoi.

Points of topoi

A point of a topos X is a geometric morphism from the topos of sets to X.

If X is an ordinary space and x is a point of X, then the functor that takes a sheaf F to its stalk F_x has a right adjoint (the "skyscraper sheaf" functor), so an ordinary point of X also determines a topos-theoretic point.

Ringed topoi

A ringed topos is a pair (X,R), where X is a topos and R is a commutative ring object in X. Most of the constructions of ringed spaces go through for ringed topoi. The category of R-module objects in X is an abelian category with enough injectives. A more useful abelian category is the subcategory of quasi-coherent R-modules: these are R-modules that admit a presentation.

Another important class of ringed topoi, besides ringed spaces, are the etale topoi of Deligne-Mumford stacks.

Homotopy theory of topoi

Michael Artin and Barry Mazur associated to any topos a pro-simplicial set. Using this inverse system of simplicial sets one may sometimes associate to a homotopy invariant in classical topology an inverse system of invariants in topos theory.

The pro-simplicial set associated to the etale topos of a scheme is a pro-finite simplicial set. Its study is called étale homotopy theory.

Elementary topoi (topoi in logic)

Introduction

A traditional axiomatic foundation of mathematics is set theory, in which all mathematical objects are ultimately represented by sets (even functions which map between sets). More recent work in category theory allows this foundation to be generalized using topoi; each topos completely defines its own mathematical framework. The category of sets forms a familiar topos, and working within this topos is equivalent to using traditional set theoretic mathematics. But one could instead choose to work with many alternative topoi. A standard formulation of the axiom of choice makes sense in any topos, and there are topoi in which it is invalid. Constructivists will be interested to work in a topos without the law of excluded middle. If symmetry under a particular group G is of importance, one can use the topos consisting of all G-sets.

It is also possible to encode an algebraic theory, such as the theory of groups, as a topos. The individual models of the theory, i.e. the groups in our example, then correspond to functors from the encoding topos to the category of sets that respect the topos structure.

Formal definition

When used for foundational work a topos will be defined axiomatically; set theory is then treated as a special case of topos theory. Building from category theory, there are multiple equivalent definitions of a topos. The following has the virtue of being concise, if not illuminating:

A topos is a category which has the following two properties:

All limits taken over finite index categories exist.
Every object has a power object.

From this one can derive that

All colimits taken over finite index categories exist.
The category has a subobject classifier.
Any two objects have an exponential object.
The category is cartesian closed.

In many applications, the role of the subobject classifier is pivotal, whereas power objects are not. Thus some definitions reverse the roles of what is defined and what is derived.

Explanation

As noted above, a topos is a category C having all finite limits and hence in particular the empty limit or final object 1. It is then natural to treat morphisms of the form x: 1 → X as elements x ∈ X. Morphisms f: X → Y then correspond to functions mapping each element x ∈ X to the element fx ∈ Y, with the caveat that two or more morphisms may correspond to the same function, that is, we cannot assume that the functor C(1,-): C → Set is faithful. Since there can only be one morphism to the domain of an element, elements are always monics.

Also noted above is that a topos has a subobject classifier Ω, an object of C incorporating an element (and hence monic) t ∈ Ω, the generic subobject, with the property that every monic m: X' → X arises as a pullback of the generic subobject along a unique morphism f: X → Ω, as per Figure 1.

Figure 1. m as a pullback of the generic subobject t along f.

But the pullback of a monic is a monic, whence every pullback of the monic t along f: X → Ω is a monic. We then have a bijection between the pullbacks of the monic t with codomain X and the monics to X, the left edge ! being the unique morphism to 1. This is the sense in which Ω is a subobject classifier: it implicitly classifies each monic m according to its associated f, and the monics thus associated to a given f constitute a subobject of X.

When C(1,-) is faithful, i.e. C is concrete, the monics m: X' → X are exactly the injections from X' to X, with each such m inducing the subset {mx| x ∈ X' } of X. The monics inducing a given such subset are exactly those comprising a subobject of X. At the same time the characteristic morphisms f: X → Ω become characteristic functions, with f⁻¹(t) denoting the set of those elements x ∈ X for which fx = t. The correspondence between subobjects and characteristic functions established by the definition of topos then reduces to the equality {mx| x ∈ X' } = f⁻¹(t) associated with the concrete notion of characteristic function.

This elementary or first-order notion of subobject in terms of classification by f corresponds to an independently derived second-order notion of subobject of X predating the concept of topos. Given two monics m, n from respectively Y and Z to X, we say that m ≤ n when there exists a morphism p: Y → Z for which np = m, inducing a preorder on monics to X. When m ≤ n and n ≤ m we say that m and n are equivalent. The subobjects of X are the equivalence classes of its monics. This explicit notion of subobject as a class is in complete agreement with that implicitly created by the subobject classifier.

Further examples

If C is a small category, then the functor category Set^C (consisting of all covariant functors from C to sets, with natural transformations as morphisms) is a topos. For instance, the category of all directed graphs is a topos. A graph consists of two sets, an arrow set and a vertex set, and two functions between those sets, assigning to every arrow its start and end vertex. The category of graphs is thus equivalent to the functor category Set^C, where C is the category with two objects joined by two morphisms.

The categories of finite sets, of finite G-sets and of finite directed graphs are also topoi.

References

John Baez: Topos theory in a nutshell, http://math.ucr.edu/home/baez/topos.html. A gentle introduction.
Stephen Vickers: Toposes pour les nuls and Toposes pour les vraiment nuls. Available at Vickers’ website. Elementary and even more elementary introductions to toposes as generalized spaces.
Illusie, Luc, "What is a ... topos?", Notices of the AMS, <http://www.ams.org/notices/200409/what-is-illusie.pdf>

The following textbooks provide easy paced first introductions (including basics of category theory). They should be suitable for students of various—even non-mathematical—disciplines:

F. William Lawvere and Stephen H. Schanuel: Conceptual Mathematics: A First Introduction to Categories, Cambridge University Press, Cambridge, 1997. An "introduction to categories for computer scientists, logicians, physicists, linguists, etc." (cited from cover text).
F. William Lawvere and Robert Rosebrugh: Sets for Mathematics, Cambridge University Press, Cambridge, 2003. Discusses the foundations of mathematics from a categorical perspective. A book "for students who are beginning the study of advanced mathematical subjects".

The original work of Grothendieck

Grothendieck and Verdier: Théorie des topos et cohomologie étale des schémas (known as SGA4)". New York/Berlin: Springer, ??. (Lecture notes in mathematics, 269–270)

Interesting research books that are provide introductions to topos theory (or to a specific aspect of it), but which do not primarily cater to students. The given order roughly (!) reflects the difficulty of the level of exposition:

Colin McLarty: Elementary Categories, Elementary Toposes, Clarendon Press, Oxford, 1992. Includes a nice introduction of the basic notions of category theory, topos theory, and topos logic. Assumes very few prerequisites.
Robert Goldblatt: Topoi, the Categorial Analysis of Logic. North-Holland, New York, 1984. (Studies in logic and the foundations of mathematics, 98.). A good start.

This book has been reprinted by Dover Publications, Inc (2006). The book can also be accessed freely on Robert Goldblatt's homepage: Topoi, the Categorial Analysis of Logic.

John L. Bell: The development of categorical logic. http://publish.uwo.ca/~jbell/catlogprime.pdf (PDF)
Saunders Mac Lane and Ieke Moerdijk: Sheaves in Geometry and Logic: a First Introduction to Topos Theory, Springer, New York, 1992. More complete, and more difficult to read.
Michael Barr and Charles Wells: Toposes, Triples and Theories, Springer, 1985. Corrected online version at http://www.cwru.edu/artsci/math/wells/pub/ttt.html. More concise than Sheaves in Geometry and Logic, but not an easy reading for the beginner.

Works which serve as a reference for experts in the field rather than as a treatment suitable for first introduction:

Francis Borceux: Handbook of Categorical Algebra 3: Categories of Sheaves, Volume 52 of the Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1994. The third part of "Borceux' remarkable magnum opus", as Johnstone has labelled it. Still suitable as an introduction, though beginners may find it hard to recognize the most relevant results among the huge amount of material given.
Peter T. Johnstone: Topos Theory, L. M. S. Monographs no. 10, Academic Press, 1977. For a long time the standard compendium on topos theory. However, it has also been described as "far too hard to read, and not for the faint-hearted", as quoted by Johnstone himself.
Peter T. Johnstone: Sketches of an Elephant: A Topos Theory Compendium, Oxford Science Publications, Oxford, 2002. Johnstone’s overwhelming compendium. As of early 2006, two of the scheduled three volumes were available.

Books that target special applications of topos theory:

Maria Cristina Pedicchio and Walter Tholen (editors): Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory. Volume 97 of the Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2004. Includes many interesting special applications.

Other encyclopedias

Topos on PlanetMath